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Advances in Mathematics 244 (2013) 441–464www.elsevier.com/locate/aim

On compact generation of deformed schemes

Wendy Lowena,∗, Michel Van den Berghb

a Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheimcampus, Middelheimlaan 1, 2020 Antwerp,Belgium

b Departement WNI, Universiteit Hasselt, 3590 Diepenbeek, Belgium

Received 10 February 2012; accepted 25 April 2013Available online 12 June 2013

Communicated by Tony Pantev

Abstract

We obtain a theorem which allows to prove compact generation of derived categories of Grothendieckcategories, based upon certain coverings by localizations. This theorem follows from an applicationof Rouquier’s cocovering theorem in the triangulated context, and it implies Neeman’s result oncompact generation of quasi-compact separated schemes. We prove an application of our theoremto non-commutative deformations of such schemes, based upon a change from Koszul complexes toChevalley–Eilenberg complexes.c⃝ 2013 Elsevier Inc. All rights reserved.

Keywords: Grothendieck category; Derived category; Compact generation; Deformation; Surface

1. Introduction

Compact generation of triangulated categories was introduced by Neeman in [16]. One ofthe motivating situations is given by derived categories of “nice” schemes (i.e. quasi-compactseparated schemes in [16], based upon the work of Thomason and Trobaugh [23], later extendedto quasi-compact quasi-separated schemes by Bondal and Van den Bergh in [2]). The ideas ofthe proofs later crystallized in Rouquier’s (co)covering theorem [21] which describes a certaincovering-by-Bousfield-localizations situation in which compact generation (later extended toα-compact generation by Murfet in [15]) of a number of “smaller pieces” entails compact

∗ Corresponding author.E-mail addresses: [email protected] (W. Lowen), [email protected] (M. Van den Bergh).

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442 W. Lowen, M. Van den Bergh / Advances in Mathematics 244 (2013) 441–464

generation of the whole triangulated category. The notions needed in the (co)covering conceptcan be interpreted as categorical versions of standard scheme constructions like unions andintersections of open subsets, and in the setup of Grothendieck categories rather than triangulatedcategories they have been important in non-commutative algebraic geometry (see e.g. [25,26,19,22]). In this paper we apply Rouquier’s theorem in order to obtain a (co)covering theoremfor Grothendieck categories based upon these notions, which can be used to prove compactgeneration of derived categories of Grothendieck categories (see Theorem 2.28 in the paper).

Theorem 1.1. Let C be a Grothendieck category with a compatible covering of affine localizingsubcategories Si ⊆ C for i ∈ I = {1, . . . , n}. Suppose:

(1) D(C/Si ) is compactly generated for every i ∈ I .(2) For every i ∈ I and ∅ = J ⊆ I \ {i}, suppose the essential image E of

j∈J

S j −→ C −→ C/Si

is such that DE (C/Si ) is compactly generated in D(C/Si ).

Then D(C) is compactly generated, and an object in D(C) is compact if and only if its image inevery D(C/Si ) for i ∈ I is compact.

When applied to the category of quasi-coherent sheaves over a quasi-compact separatedscheme, the theorem implies Neeman’s original result.

Our interest in the intermediate Theorem 1.1 comes from its applicability to Grothendieckcategories that originate as “non-commutative deformations” of schemes, more precisely abeliandeformations of categories of quasi-coherent sheaves in the sense of [14], which are shownin [13] to arise from deformations as algebroid prestacks. After formulating a general resultfor deformations (Theorem 3.8), based upon lifting compact generators under deformation, wespecialize further to the scheme case in Theorem 5.2. We use the description of deformationsfrom [13] using non-commutative twisted presheaf deformations of the structure sheaf on anaffine open cover.

When all involved deformed rings are commutative, using liftability of Koszul complexesunder deformation, the corresponding twisted deformations are seen to be compactly generated,a fact which also follows from [24]. In our main Theorem 1.2 (see Theorem 5.10 in the paper),we show that actually all non-commutative deformations are compactly generated.

Theorem 1.2. Let X be a quasi-compact separated scheme over a field k. Then every flatdeformation of the abelian category Qch(X) over a finite dimensional commutative local k-algebra has a compactly generated derived category.

The proof is based upon the following lifting result for Koszul complexes (see Theorem 4.3in the paper):

Theorem 1.3. Let A be a commutative k-algebra and f = ( f1, . . . , fn) a sequence of elementsin A. For d ≥ 1 there exists a perfect complex Xd ∈ D(A) generating the same thick subcategoryof D(A) as the Koszul complex K ( f ) and satisfying the following property: let (R,m) be a finitedimensional commutative local k-algebra with md

= 0 and R/m = k. Then Xd may be lifted toany R-deformation of A.

W. Lowen, M. Van den Bergh / Advances in Mathematics 244 (2013) 441–464 443

Concretely, let n be the Lie algebra freely generated by x1, . . . , xn subject to the relationsthat all expressions involving ≥d brackets vanish. Sending xi to fi makes A into a right n-representation. Then Xd is defined as the Chevalley–Eilenberg complex (A ⊗k ∧n, dC E ) of A.Clearly X1 = K ( f ). It appears that in general one should think of Xd as a kind of “higher Koszulcomplex”.

2. Coverings of Grothendieck categories

Localization theory of abelian categories and Grothendieck categories goes back to the workof Gabriel [6], which actually contains some of the important seeds of non-commutative algebraicgeometry, like the fact that noetherian (this condition was later eliminated in the work ofRosenberg [20]) schemes can be reconstructed from their abelian category of quasi-coherentsheaves. In the general philosophy (due to Artin, Tate, Stafford, Van den Bergh and others)that non-commutative schemes can be represented by Grothendieck categories “resembling”quasi-coherent sheaf categories, localizations of such categories have been a key ingredient inthe development of the subject by Rosenberg, Smith, Van Oystaeyen, Verschoren, and others(see eg. [19,22,25,26]). In particular, Van Oystaeyen and Verschoren investigated a notion ofcompatibility between localizations (see [25,3]).

More recent approaches to non-commutative algebraic geometry (due to Bondal, Kontsevich,Toen and others) take triangulated categories (and algebraic enhancements like dg or A∞

algebras and categories) as models for non-commutative spaces. The beautiful abelianlocalization theory was paralleled by an equally beautiful triangulated localization theory,based upon Verdier and Bousfield localization, see e.g. [10] and the references therein. Byconsidering appropriate unbounded derived categories, every Grothendieck localization givesrise to a Bousfield localization.

Recently, the notion of properly intersecting Bousfield subcategories was introduced byRouquier in the context of his cocovering theorem concerning compact generation of certaintriangulated categories [21]. The condition bears a striking similarity to the notion ofcompatibility in the Grothendieck context, which is even reinforced by the characterizationproved by Murfet in [15].

In this section, we introduce all the relevant notions in both contexts, and we observe thatin the special situation where the right adjoints of a collection of compatible localizations ofGrothendieck categories are exact, they give rise to properly intersecting Bousfield localizations,and Grothendieck coverings give rise to triangulated coverings. We go on to deduce a coveringtheorem for Grothendieck categories (Theorem 2.28) which allows to prove compact generationof the derived category.

2.1. Coverings of abelian categories

We first review the situation for abelian categories. Let C be an abelian category. A localizationof C consists of an exact functor

a : C −→ C′

with a fully faithful right adjoint i : C′−→ C. A subcategory S ⊆ C is called a Serre subcategory

if it is closed under subquotients and extensions. A Serre subcategory gives rise to an exactGabriel quotient

a : C −→ C/S


with Ker(a) = S . The Serre subcategory S is called localizing if a is the left adjoint in alocalization.

Now suppose C is Grothendieck. Then S is localizing precisely when S is moreover closedunder coproducts. Conversely, for every localization a : C −→ C′, S = Ker(a) is localizing, afactors over an equivalence C/S ∼= C′, and putting

S ⊥= {C ∈ C | HomC(S,C) = 0 = Ext1C(S,C)∀S ∈ S},

the right adjoint i : C′−→ C factors over an equivalence C′ ∼= S ⊥. Clearly, in the picture of a

localization, the data of S , a and i determine each other uniquely.Let C be an abelian category. For full subcategories S1, S2 of C, the Gabriel product is given

by

S1 ∗ S2 = {C ∈ C | ∃ S1 ∈ S1, S2 ∈ S2, 0 −→ S1 −→ C −→ S2 −→ 0}.

Clearly, if S contains the zero object, then S is closed under extensions if and only if S ∗ S = S .An easy diagram argument reveals that the Gabriel product is associative.

Definition 2.1 ([25,3]). Full subcategories S1, S2 of C are called compatible if

S1 ∗ S2 = S2 ∗ S1.

Proposition 2.2 ([25,26]). Consider localizations (S1, a1, i1) and (S2, a2, i2) of C. Put q1 =

i1a1 and q2 = i2a2. The following are equivalent:(1) S1 and S2 are compatible.(2) q1(S2) ⊆ S2 and q2(S1) ⊆ S1.(3) q1q2 = q2q1.

In the situation of Proposition 2.2, we speak about compatible localizations. A collection ofSerre subcategories (or localizations) is called compatible if the corresponding localizations arepairwise compatible. For two compatible Serre subcategories S1 and S2, S1 ∗ S2 is the smallestSerre subcategory containing S1 and S2. If S1 and S2 are localizing, then so is S1 ∗ S2.

Definition 2.3. A collection Σ of Serre subcategories of C is called a covering of C ifΣ =

S∈Σ

S = 0.

By this definition, the collection of functors a : C −→ C/S with S ∈ Σ “generates” C in thesense that C ∈ C is non-zero if and only if a(C) is non-zero for some S ∈ Σ .

Proposition 2.4. Consider a collection Σ of localizing Serre subcategories of C. The followingare equivalent:(1) Σ is a covering of C.(2) The objects i(D) for D ∈ C/S and S ∈ Σ cogenerate C, i.e. a morphism C ′

−→ C in C isnon-zero if and only if there exists a morphism C −→ i(D) with D ∈ C/S for some S ∈ Σsuch that C ′

−→ C −→ i(D) is non-zero.

Proof. This easily follows from the adjunction between a and i . �

The notion of a compatible covering is inspired by open coverings of schemes. For a coveringcollection j : Ui −→ X of open subschemes of a scheme X (i.e. X = ∪Ui ), the collection oflocalizations j∗ : Qch(X) −→ Qch(Ui ) constitutes a compatible covering of Qch(X).


2.2. Descent categories

Consider a compatible collection Σ of localizations C j of C indexed by a finite set I . We thenobtain commutative (up to natural isomorphism) diagrams of localizations

Cak //

a j

��

Ck

akk j

��C j

a jk j

// Ck j

with Ck = S ⊥

k , Ck j = (Sk ∗ S j )⊥

= C j ∩ Ck .Using associativity of the Gabriel product and compatibility of the localizations, we obtain

for each J = { j1, . . . , jp} ⊆ I a localizing subcategory

S J = S j1 ∗ · · · ∗ S jp

of C with corresponding localization

C J = C j1 ∩ · · · ∩ C jp

of C, and all the S J are compatible. In particular, we obtain for every inclusion K ⊆ J a furtherlocalization aK

J : CK −→ C J left adjoint to the inclusion i KJ : C J −→ CK . It is easily seen that

for K ⊆ J1 and K ⊆ J2, the localizations aKJ1

and aKJ2

are compatible. Let ∆∅ be the categoryof finite subsets of I ordered by inclusions, and let ∆ be the subcategory of non-empty subsets.Putting C∅ = C, the categories C J for J ⊆ I can be organized into a pseudofunctor

C• : ∆∅ −→ Cat : J −→ C J

with the aKJ for K ⊆ J as restriction functors. Hence, C• is a fibered category of localizations in

the sense of [13, Definition 2.4].We define the descent category Des(Σ ) of Σ to be the descent category Des(C•|∆), i.e. it is a

bi-limit of the restricted pseudofunctor C•|∆.In particular, we obtain a natural comparison functor

C −→ Des(Σ ).

We conclude:

Proposition 2.5. The compatible collection Σ of localizations of C is a covering if and only ifthe comparison functor C −→ Des(Σ ) is faithful.

Conversely, descent categories yield a natural way of constructing an abelian category coveredby a given collection of abelian categories. More precisely, let I be a finite index set, let ∆ be asabove, and let

C• : ∆ −→ Cat : J −→ C J

be a pseudofunctor for which every aKJ : CK −→ C J for K ⊆ J is a localization with

right adjoint i KJ and Ker(aK

J ) = S KJ . Suppose moreover that for K ⊆ J1 and K ⊆ J2

the corresponding localizations are compatible and S KJ1∪J2

= S KJ1

∗ S KJ2

. Consider the descent


category Des(C•) with canonical functors

aK : Des(C•) −→ CK : (X J )J −→ X K .

Proposition 2.6. (1) The functor aK is a localization with fully faithful right adjoint iK with

aJ iK = i JK∪J aK

K∪J : CK −→ C J .

(2) The localizations aK are compatible.(3) The localizations aK constitute a covering of Des(C•).(4) If the functors i J

K∪J are exact, then so are the functors iK .

Proof. (1) For X ∈ CK , using compatibility of the localizations occurring in C•,(i J

K∪J aKK∪J (X))J can be made into a descent datum iK (X), and iK can be made into a functor

right adjoint to aK . Since aK iK = 1CK , the functor iK is fully faithful. (2), (3) Immediate fromProposition 2.5. (4) Immediate from the formula in (1). �

2.3. Coverings of triangulated categories

Next we review the situation for triangulated categories. For an excellent introduction to thelocalization theory of triangulated categories, we refer the reader to [10].

Let T be a triangulated category. A (Bousfield) localization of T consists of an exact functor

a : T −→ T ′

with a fully faithful (automatically exact) right adjoint i : T ′−→ T . A subcategory I ⊆ T is

called triangulated if it is closed under cones and shifts and thick if it is moreover closed underdirect summands. A thick subcategory gives rise to an exact Verdier quotient

a : T −→ T /I

with Ker(a) = I . The thick subcategory I is called a Bousfield subcategory if a is the left adjointin a localization.

For every localization a : T −→ T ′, I = Ker(a) is Bousfield, a factors over an equivalenceT /I ∼= T ′, and putting

I ⊥= {T ∈ T | HomT (I, T ) = 0 ∀I ∈ I},

the right adjoint i : T ′−→ T factors over an equivalence T ′ ∼= I ⊥. Clearly, in the picture of a

localization, the data of I , a and i determine each other uniquely.For full subcategories I1, I2 of T , the Verdier product is given by

I1 ∗ I2 = {T ∈ T | ∃ I1 ∈ I1, I2 ∈ I2, I1 −→ T −→ I2 −→}.

Clearly, if I contains the zero object and is closed under shifts, then I is triangulated if and onlyif I = I ∗ I .

Definition 2.7 ([21,15]). Full subcategories I1, I2 of T are called compatible if

I1 ∗ I2 = I2 ∗ I1.

Remark 2.8. In Definition 2.7, we adopted the same terminology as in the abelian setup, seeDefinition 2.1. Note that in our Refs. [21,15], compatible subcategories are called properlyintersecting.


For two compatible triangulated subcategories I1 and I2, I1 ∗ I2 is the smallest triangulatedsubcategory containing I1 and I2. If I1 and I2 are localizing (resp. Bousfield), the same holdsfor I1 ∗ I2 (see [21,15]).

Proposition 2.9 ([21,15]). Consider localizations (I1, a1, i1) and (I2, a2, i2) of C. Put q1 =

i1a1 and q2 = i2a2. The following are equivalent:

(1) I1 and I2 intersect properly.(2) q1(I2) ⊆ I2 and q2(I1) ⊆ I1.(3) q1q2 = q2q1.

In the situation of Proposition 2.9, we speak about compatible localizations. A collection ofsubcategories (resp. localizations) is called compatible if the subcategories (resp. localizations)are pairwise compatible.

Definition 2.10. A collection Θ of full subcategories of T is called a covering of T ifΘ =

I∈Θ

I = 0.

Remark 2.11. In [21], the term cocovering is reserved for a collection of Bousfield subcategorieswhich is covering in the sense of Definition 2.10 and properly intersecting.

By Definition 2.10, for a covering collection Θ of thick subcategories, the collection ofquotient functors a : T −→ T /I with I ∈ Θ “generates” T in the sense that T ∈ T isnon-zero if and only if a(T ) is non-zero for some I ∈ Θ .

Proposition 2.12. Consider a collection Θ of Bousfield subcategories of T . The following areequivalent:

(1) Θ is a covering of T .(2) The objects i(D) for D ∈ T /I and I ∈ Θ cogenerate T , i.e. an object T in T is non-zero if

and only if there exists a non-zero morphism T −→ i(D) with D ∈ T /I for some I ∈ Θ .

Proof. This easily follows from the adjunction between a and i . �

2.4. Induced coverings

Given the formal parallelism between Sections 2.1 and 2.3, and the fact that a localizationa : C −→ D with right adjoint i : D −→ C of Grothendieck categories gives rise to an inducedBousfield localization

La = a : D(C) −→ D(D)

with fully faithful right adjoint

Ri : D(D) −→ D(C)

of the corresponding derived categories, it is natural to ask what happens to the notions ofcompatibility and coverings under this operation of taking derived categories.

For coverings, the situation is very simple. For a full subcategory S of C, let DS(C) denotethe full subcategory of D(C) consisting of complexes whose cohomology lies in S .


Lemma 2.13. Let a : C −→ D be an exact functor between Grothendieck categories withKer(a) = S , and consider La = a : D(C) −→ D(D). We have Ker(La) = DS(C).

Proof. For a complex X ∈ D(C), we have Hn(a(X)) = a(Hn(X)). �

Lemma 2.14. For a collection Σ of full subcategories of a Grothendieck category C, we have

D S∈Σ

S(C) =

S∈Σ

DS(C).

Proposition 2.15. Let Σ be a collection of full subcategories of a Grothendieck category C. ThenΣ is a covering of C if and only if the collection {DS(C) | S ∈ Σ } is a covering of D(C).

Now consider a Grothendieck category C and localizations ak : C −→ Dk with right adjointsik , qk = ikak , and Sk = Ker(ak) for k ∈ {1, 2}.

Taking derived functors yields Bousfield localizations Lak = ak : D(C) −→ D(C/Sk) withright adjoints Rik : D(C/Sk) −→ D(C) and Ker(Lak) = DSk (C).

We have the following inclusion between thick subcategories:

Lemma 2.16. We have DS1(C) ∗ DS2(C) ⊆ DS1∗S2(C).

Proof. A triangle X1 −→ X −→ X2 −→ with Xk ∈ DSk (C) gives rise to a long exact sequence· · · −→ Hn X1 −→ Hn X −→ Hn X2 −→ · · · . Since Sk is closed under subquotients, weobtain an exact sequence 0 −→ S1 −→ Hn X −→ S2 −→ 0 with Sk ∈ Sk . �

In general, we have:

Proposition 2.17. If DS1(C) and DS2(C) are compatible in D(C), then S1 and S2 arecompatible in C.

Proof. Immediate from Lemma 2.18 and the characterizations (2) in Propositions 2.2 and 2.9.�

Lemma 2.18. If Ri1a1(DS2(C)) ⊆ DS2(C), then i1a1(S2) ⊆ S2.

Proof. Let S2 ∈ S2. We have i1a1(S2) = R0i1a1(S2) = H0 Ri1a1(S2) and since Ri1a1(S2) ∈

DS2(C), it follows that i1a1(S2) ∈ S2 as desired. �

Unfortunately, the converse implication does not hold in general. However, in the specialsituation where i1 and i2 are exact, it is equally straightforward.

Definition 2.19. A localization a : C −→ D is called affine if the right adjoint i : D −→ Cis exact. A localizing subcategory S ⊆ C is called affine if the corresponding localizationC −→ C/S is affine.

Example 2.20. If X is a quasi-compact separated scheme and U ⊂ X an affine open subscheme,then Qch(X) −→ Qch(U ) is an affine localization.

Remark 2.21. Other variants of “affineness” have been used in the non-commutative algebraicgeometry literature. For instance, Paul Smith calls an inclusion functor i : D −→ C affine if ithas both adjoints. In particular, if i : D −→ C is the right adjoint in a localization, it becomes aforteriori exact. In this paper, we have chosen the most convenient notion of “affineness” for ourpurposes.


Proposition 2.22. If S1 and S2 are compatible and affine, then DS1(C) and DS2(C) arecompatible.

Proof. Immediate from Lemma 2.23 and the characterizations (2) in Propositions 2.2 and2.9. �

Lemma 2.23. If i1a1(S2) ⊆ S2 and i1 is exact, then Ri1a1(DS2(C)) ⊆ DS2(C).

Proof. For a complex X ∈ DS2(C), we have Ri1a1(X) = i1a1(X) and since i1a1 is exact,Hn(i1a1(X)) = i1a1(Hn(X)) ∈ S2. �

The affineness condition in Proposition 2.22 does not describe the only situation wherecompatible abelian localizations give rise to compatible Bousfield localizations, but it is the onlysituation we will need in this paper. To end this section, we will describe another situation,inspired by the behavior of large categories of sheaves of modules.

Recall that the functor i : C/S −→ C has finite cohomological dimension if there exists anN ∈ Z such that if X ∈ D(C/S) has Hn(X) = 0 for n > 0, then Rni(X) = Hn(Ri(X)) = 0 forn ≥ N .

Proposition 2.24. Let S1 and S2 be compatible and suppose the following conditions hold:

(1) There exist a class of objects A ⊆ C and classes Ak ⊆ C/Sk consisting of ik-acyclic objectssuch that ak(A) ⊆ Ak and ik(Ak) ⊆ A.

(2) The functors ik have finite cohomological dimension.

Then DS1(C) and DS2(C) are compatible.

Example 2.25. Let X be a quasi-compact scheme with quasi-compact open subschemes j1 :

U1 −→ X and j2 : U2 −→ X . We have restriction functors j∗k : Mod(X) −→ Mod(Uk)

between the categories of all sheaves of modules with right adjoints ik,∗ : Mod(Uk) −→

Mod(X) with finite cohomological dimension. In Proposition 2.24, we can take for A and Akthe classes of flabby sheaves. Hence the localizations j∗k : D(Mod(X)) −→ D(Mod(Uk)) arecompatible.

2.5. Rouquier’s theorem

Compactly generated triangulated categories were invented by Neeman [16] with the compactgeneration of derived categories of “nice” schemes as one of the principal motivations. As provedin [16], for a scheme with a collection of ample invertible sheaves, these sheaves constitute acollection of compact generators of the derived category. But also in [16], a totally differentproof of compact generation is given for arbitrary quasi-compact separated schemes, based uponthe work of Thomason and Trobaugh [23]. The result is further improved by Bondal and Van denBergh in [2], where a single compact generator is constructed for quasi-compact semi-separatedschemes. These proofs are by induction on the opens in a finite affine cover, and the ingredientseventually crystallized in Rouquier’s theorem [21] which is entirely expressed in terms of acover of a triangulated category. Finally, in [15], Murfet obtained a version of the theorem withcompactness replaced by α-compactness. We start by recalling the theorem.

Theorem 2.26 ([15]). Let T be a triangulated category with small coproducts with a compatiblecovering of Bousfield subcategories Ii ⊆ T for i ∈ I = {1, . . . , n}. Let α be a regular cardinal.Suppose:


(1) T /Ii is α-compactly generated for every i ∈ I .(2) For every i ∈ I and ∅ = J ⊆ I \ {i}, the essential image of

j∈J

I j −→ T −→ T /Ii

is α-compactly generated in T /Ii .

Then T is α-compactly generated, and an object in T is α-compact if and only if its image inevery T /Ii for i ∈ I is α-compact.

Remark 2.27. The α = ℵ0-case of the theorem is Rouquier’s cocovering theorem [21].

We now obtain the following application to Grothendieck categories:

Theorem 2.28. Let C be a Grothendieck category with a compatible covering of affine localizingsubcategories Si ⊆ C for i ∈ I = {1, . . . , n}. Suppose:

1. D(C/Si ) is α-compactly generated for every i ∈ I .2. For every i ∈ I and ∅ = J ⊆ I \ {i}, suppose the essential image E of

j∈J

S j −→ C −→ C/Si

is such that DE (C/Si ) is α-compactly generated in D(C/Si ).

Then D(C) is α-compactly generated, and an object in D(C) is α-compact if and only if its imagein every D(C/Si ) for i ∈ I is α-compact.

Proof. This is an application of Theorem 2.26 by invoking Propositions 2.15 and 2.22 andLemma 2.29. �

Lemma 2.29. With the notations of Theorem 2.28, ∩ j∈J S j is a localizing Serre subcategorywhich is compatible with Si , and the essential image E of

j∈J

S j −→ C −→ C/Si

is a localizing Serre subcategory given by the kernel of

C/Si −→ C/(Si ∗

j∈J

S j ).

The essential image ofj∈J

DS j (C) −→ D(C) −→ D(C/Si )

is given by DE (C/Si ).

Remark 2.30. By the Gabriel–Popescu theorem, all Grothendieck categories are localizationsof module categories, and thus their derived categories are well-generated [18,9] (and thus α-compactly generated for some α) as localizations of compactly generated derived categories ofrings. However, they are not necessarily compactly generated as was shown in [17].

Remark 2.31. Compatibility between localizations can be considered a commutative phe-nomenon (after all, it expresses that two localization functors commute). The non-commutative


topology developed by Van Oystaeyen [25] encompasses notions of coverings (and in fact, non-commutative Grothendieck topologies) which apply to the situation of non-commuting local-izations. An investigation whether this approach can be extended to the triangulated setup, andwhether it is possible to obtain results on compact generation extending Theorems 2.26 and 2.28,is work in progress.

3. Deformations

In this section we obtain an application of Theorem 2.28 to deformations of Grothendieckcategories, based upon application to the undeformed categories (Theorem 3.8). For simplicity,we focus on compact generation (α = ℵ0). By the work of Keller [7], compact generation ofthe derived category D(C) of a Grothendieck category leads to the existence of a dg algebra A– the derived endomorphism algebra of a generator – representing the category in the sense thatD(C) ∼= D(A). At this point, most of non-commutative derived algebraic geometry has beendeveloped with dg algebras (or A∞-algebras) as models, although a definitive theory shouldalso include more general algebraic enhancements on the level of the entire categories. For thetopic of deformations, a satisfactory treatment on the level of dg algebras does certainly notexist in complete generality [8], due to obstructions which also play an important role in thepresent paper. A deformation theory for triangulated categories on the level of enhancements ofthe entire categories is still under construction [11,4], and is also subject to obstructions. Thus,Grothendieck enhancements are the only ones for which a satisfactory intrinsic deformationtheory exists for the moment, and for this reason our intermediate Theorem 2.28 is crucial.

3.1. Deformation and localization

Infinitesimal deformations of abelian categories were introduced in [14]. We deform along asurjective ringmap R −→ k between coherent commutative rings, with a nilpotent kernel I andsuch that k is finitely presented over R. This includes the classical infinitesimal deformation setupin the direction of Artin local k-algebras. Deformations are required to be flat in an appropriatesense, which was introduced in [14]. It was shown in the same paper that deformationsof Grothendieck categories remain Grothendieck. The interaction between deformation andlocalization was treated in [14, Section 7].

Let C −→ D be a deformation of Grothendieck categories. There are inverse bijectionsbetween the Serre subcategories of C and the Serre subcategories of D described by the maps

S −→ S = ⟨S⟩D = {D ∈ D | k ⊗R D ∈ S}

and

S −→ S ∩ C.

These restrict to bijections between localizing subcategories, and for corresponding localizingsubcategories S of C and S of D, there is an induced deformation C/S −→ D/S and there arecommutative diagrams

D a // D/S

C

OO

a// C/S

OO D D/Sıoo

C

OO

C/S.

OO

ioo

(1)


Proposition 3.1. Let Σ be a collection of Serre subcategories of C and consider thecorresponding collection Σ = {S | S ∈ Σ } of Serre subcategories of D. Then Σ is a covering ofC if and only if Σ is a covering of D.

Proof. Immediate from Lemma 3.2. �

Lemma 3.2. Let Σ be a collection of Serre subcategories of C. We haveS∈Σ

S =

S∈Σ

S.

Proof. Immediate from the description of the bijections between Serre subcategories of C andD. �

Proposition 3.3 ([13, Proposition 3.8]). Let Sk ⊆ C be localizing subcategories for k ∈ {1, 2}.If S1 and S2 are compatible in C, then S1 and S2 are compatible in D. In this case, we haveS1 ∗ S2 = S1 ∗ S2.

We will need the following lifting result later on.

Lemma 3.4. Let Sk ⊆ C be compatible localizing subcategories for k ∈ {1, 2}. The essentialimage E of

S2 −→ C −→ C/S1

is the kernel of

a12 : C/S1 −→ C/S1 ∗ S2.

The lift E of E to D/S1 is the essential image of

S2 −→ D −→ D/S1.

Finally, we investigate the behavior of affine localizations in the sense of Definition 2.19 underdeformation. We use the notations from diagram (1).

Proposition 3.5. (i) There is a commutative diagram

D(D) D(D/S)Rıoo

D(C)

OO

D(C/S).

OO

Rioo

(ii) If S is an affine localizing subcategory of C, then S is an affine localizing subcategory of D.

Proof. (i) By Lemma 3.6, the commutativity of the diagram follows from the commutativityof the right hand diagram in (1), by computing the right derived functor of the composition intwo ways. (ii) follows in the standard way from (i) by writing an arbitrary object in D/S as anextension of objects in C/S . �

Lemma 3.6. The inclusion functor C/S −→ D/S sends injectives to ı -acyclic objects.


Proof. Let F ∈ D/S be the unique, D/S -injective lift of E along HomR(k,−) : D/S −→ C/S(see [14, Corollary 6.15] and [12, Proposition 5.5]). We thus have

HomR(k, F) = E .

Choose a resolution of k by finitely generated free R-modules.

· · · −→ P1 −→ P0 −→ k −→ 0. (2)

Apply HomR(−, F) to this resolution. Since injectives are coflat (all categories are assumed tobe flat) we obtain an injective resolution of E in D/S .

0 −→ E −→ HomR(P0, F) −→ HomR(P1, F) −→ · · · .

Applying ı we obtain a complex

0 −→ ı HomR(P0, F) −→ ı HomR(P1, F) −→ · · · (3)

which computes Ri ı(E).On the other hand we may also apply HomR(−, ı F) to (2) which yields an exact sequence

HomR(P0, ı F) −→ HomR(P1, ı F) −→ · · ·

which is easily seen to be the same as (3) (except for the zero on the left). The result follows. �

3.2. Lifts of compact generators

Let ι : C −→ D be a deformation of Grothendieck categories, let S be a localizing Serresubcategory of C and let S be the corresponding localizing subcategory of D.

For an abelian category A, let Ind(A) be the ind-completion of A, i.e. the closure of A insideMod(A) under filtered colimits, and let Pro(A) = (Ind(Aop

))op

be the pro-completion of A.Consider the commutative diagram

DS(D) // D(D)

DS(C)

OO

// D(C)

Rι

OO

and the derived functor

k ⊗LR − : D(Pro(D)) −→ D(Pro(C)).

For a collection A of objects in a triangulated category T , we denote by ⟨A⟩T the smallestlocalizing (i.e. triangulated and closed under direct sums) subcategory of T containing A.

Recall from [5] that we have a balanced action

− ⊗LR − : D−(Mod(R))⊗ D−(D) −→ D−(D).

The following is a refinement of [5, Proposition 5.9].

Proposition 3.7. Consider a collection g of objects of D−(D) such that the collection k ⊗LR g =

{k ⊗LR G | G ∈ g} compactly generates DS(C) inside D(C). Then g compactly generates DS(D)

inside D(D).


Proof. The objects of g are compact by [5, Proposition 5.8]. Consider ⟨g⟩D(D), i.e. the closureof g in D(D) under cones, shifts and direct dums. We are to show that ⟨g⟩D(D) = DS(D). Firstwe have to make sure that every G ∈ g is contained in DS(D). Writing I = Ker(R −→ k)as a homotopy colimit of cones of finite free k-modules, we obtain that both k ⊗

LR G and

I ⊗LR G ∼= I ⊗

Lk (k ⊗

LR D) belong to DS(C) and hence to DS(D). From the triangle

I ⊗LR G −→ G −→ k ⊗

LR G −→

we deduce that G also belongs to DS(D). Consequently ⟨g⟩D(D) ⊆ DS(D).Next we look at the other inclusion DS(D) ⊆ ⟨g⟩D(D). For an arbitrary complex D ∈ DS(D),

we can write D = hocolim∞

n=0τ≤n D with τ≤n D ∈ D−

S (D). Consequently, it suffices to show

that D−

S (D) ⊆ ⟨g⟩D(D). For D ∈ D−

S (D), consider the triangle

I ⊗LR D −→ D −→ k ⊗

LR D −→ .

First note that writing k as a homotopy colimit of cones of finite free R-modules, we see that

k ⊗LR D ∈ ⟨D⟩D(D). (4)

Using I ⊗LR D ∼= I ⊗

Lk (k ⊗

LR D), we deduce from balancedness of the derived tensor product

that I ⊗LR D and k ⊗

LR D belong to both D(C) and DS(D), whence to DS(C). Consequently, it

suffices to show that ⟨k ⊗LR g⟩D(C) = DS(C) ⊆ ⟨g⟩D(D). To see this, it suffices to consider (4)

for all D ∈ g. �

3.3. Compact generation of deformations

Putting together all our results so far, we now describe a situation in which one obtainscompact generation of the derived category D(D) of a deformation D. Let C be a Grothendieckabelian category with a deformation ι : C −→ D. Let Si ⊆ C for i ∈ I = {1, . . . , n} be acovering collection of compatible affine localizing subcategories of C and let Si ⊆ D be thecorresponding covering collection of compatible affine localizing subcategories of D.

Theorem 3.8. Suppose:

(1) For every i ∈ I , there is a collection gi of objects of D−(D/Si ) such that the collectionk ⊗

LR gi compactly generates D(C/Si ).

(2) For every i ∈ I and J ⊆ I \ {i}, the essential image E ofj∈J

S j −→ C −→ C/Si

is such that there is a collection g of objects of D−(D/Si ) for which the collection k ⊗LR g

compactly generates DE (C/Si ) inside D(C/Si ).

Then D(D) is compactly generated and an object in D(D) is compact if and only if its image ineach D(D/Si ) is compact.

Proof. By Propositions 3.1, 3.3 and 3.5, we are in the basic setup of Theorem 2.28. ByProposition 3.7, the collections gi and g in assumptions (1) and (2) constitute collections ofcompact generators of D(D/Si ) and of DE (D/S i ) inside D(D/S i ) respectively. Finally, usingLemma 3.4, assumptions (1) and (2) in Theorem 2.28 are fulfilled and the theorem applies to thedeformed situation. �


4. Lifting Koszul complexes

For an object M in D(A) for a k-algebra A, we denote by ⟨M⟩A the smallest thick subcategoryof D(A) containing M and by ⟨M⟩A the smallest localizing subcategory of D(A) containing M .

For a k-algebra map A −→ B we have a restriction functor

A(−) : D(B) −→ D(A)

with left adjoint given by the derived tensor product

B ⊗LA − : D(A) −→ D(B)

(the actual map A −→ B will never be in doubt).

4.1. An auxiliary result

Let k be a field and let n be the Lie algebra freely generated by x1, . . . , xn subject to therelations that all expressions involving ≥d brackets vanish. Since there are only a finite numberof expressions in (xi )i involving <d brackets, n is finite dimensional over k. Let U be theuniversal enveloping algebra of n. Let I ⊆ U be the two sided ideal generated by ([xi , x j ])i j .Then U/I = k[x1, . . . , xn]. Our arguments below will be mostly based on the k-algebra maps

U // U/Ixi →0 // k.

The left k-module k gives rise to a left U -module U k = U/(x1, . . . , xn) and a left U/I -module U/I k = (U/I )/(x1, . . . , xn). Since U (as well as U/I ) is noetherian of finite globaldimension, U k is a perfect left U -module and U/I k is a perfect U/I -module. By tensoring weobtain another perfect U/I -module: U/I ⊗

LU U k.

The following result was pointed out to us by the referee:

Lemma 4.1. Consider A = k[x1, . . . , xn] and k = A/(x1, . . . , xn) ∈ D(A). For 0 = F ∈ ⟨k⟩A,we have ⟨k⟩A = ⟨F⟩A.

Proof. We are to prove that k ∈ ⟨F⟩A. Since k is perfect over A, we have k ⊗LA F ∈ ⟨F⟩A.

Further, using the ringmap A −→ k : xi −→ 0, we can view k ⊗LA F = Ak ⊗

LA F as the image

of kk ⊗LA F ∈ D(k) along the forgetful functor. But in D(k), we have kk ⊗

LA F = ⊕i Σ ni kk, and

hence Ak ⊗LA F = ⊕i Σ ni Ak. Thus k ∈ ⟨k ⊗

LA F⟩A ⊆ ⟨F⟩A as desired. �

Proposition 4.2. We have the following equalities: ⟨U/I k⟩U/I = ⟨U/I ⊗LU U k⟩U/I and

⟨U/I k⟩U/I = ⟨U/I ⊗LU U k⟩U/I .

Proof. By Lemma 4.1, it remains to prove that U/I ⊗LU U k ∈ ⟨U/I k⟩U/I . Note that ⟨U/I k⟩U/I

consists of all complexes of U/I -modules whose total cohomology is finite dimensional andreceives a nilpotent action from all the xi . To check that the cohomology of U/I ⊗

LU U k

is finite dimensional, we only have to look at the underlying k-module. Hence we computek(U/I ⊗

LU U k) (restriction for k ↩→ U/I ) which we can do using a finite free resolution of

(U/I )U as a right U -module. Things then reduce to the trivial fact k(U ⊗LU U k) ∼= k. Giving

every xi ∈ U/I degree 1, the cohomology of U/I ⊗LU U k further becomes Z-graded. Since it is

finite dimensional, the action of the xi is necessarily nilpotent. �


4.2. Koszul precomplexes

Let A be a possibly non-commutative k-algebra and consider a finite sequence of elementsx = (x1, . . . , xn) in A. We will work in the category Mod(A) of left A-modules. We define a

precomplex K (x) of A-modules with K (x)p = A ⊗k Λpkn the free A-module of rank

np

with

basis ei1 ∧ · · · ∧ ei p with i1 < · · · < i p. We define the A-linear morphism

dp : K (x)p −→ K (x)p−1

by

dp(ei1 ∧ · · · ∧ ei p ) =

pk=1

(−1)k+1xik ei1 ∧ · · · ∧ eik ∧ · · · ∧ ei p .

The differential may be compactly written as d =

i Rxi δ/δei where we consider the ei asodd and Rxi (a) = axi which yields:

d2=

1≤i< j≤p

R[x j ,xi ] δ2/δeiδe j .

Thus K (x) is a complex if and only if the (xi )i commute.

4.3. Lifting Koszul complexes

Let (R,m) be a finite dimensional k-algebra with md= 0 and R/m = k and let A′ be an

R-algebra with A′/m A′= A. Consider a sequence f = ( f1, . . . , fn) of element in A and a

sequence f ′= ( f ′

1, . . . , f ′n) of elements in A′ such that the reduction of f ′

i to A equals fi . LetK ( f ) be the Koszul complex associated to f . As soon as some of the f ′

i do not commute, theKoszul precomplex K ( f ′) fails to be a complex according to Section 4.2. For this reason, wewill now use the result of Section 4.1 to lift a perfect complex generating the same localizingsubcategory as K ( f ). In fact, this “liftable complex” happens to be independent of A′ or R! Itssize depends however in a major way on d .

Theorem 4.3. Let (R,m) be a finite dimensional algebra with md= 0 and R/m = k, and let A

be a commutative k-algebra and f = ( f1, . . . , fn) a sequence of elements in A. There exists aperfect complex X ∈ D(A) with ⟨K ( f )⟩A = ⟨X⟩A and

⟨K ( f )⟩A = ⟨X⟩A

which is such that for every R-algebra A′ with A′/m = A there exists a perfect complexX ′

∈ D(A′) with A ⊗LA′ X ′

= X. We can take X = A ⊗LU U k.

Proof. Let f ′= ( f ′

1, . . . , f ′n) be an arbitrary sequence of elements in A′ such that the reduction

of f ′

i to A equals fi . From the definition of the algebra U in Section 4.1, we obtain a commutativediagram

U //

��

A′

��U/I // A


with the horizontal maps determined by xi −→ f ′

i and xi −→ fi respectively. We thus have

A ⊗LA′(A′

⊗LU U k) = A ⊗

LU/I (U/I ⊗

LU U k). (5)

By Proposition 4.2 we have ⟨U/I k⟩U/I = ⟨U/I ⊗LU U k⟩U/I and hence

⟨A ⊗LU/I U/I k⟩A = ⟨A ⊗

LU/I (U/I ⊗

LU U k)⟩A. (6)

Over U/I = k[x1, . . . , xn], the Koszul complex K (x1, . . . , xn) constitutes a projectiveresolution of U/I k. Hence, on the left hand side of (6) we have A ⊗

LU/I U/I k =

A ⊗U/I K (x1, . . . , xn) = K ( f ). Hence, by (5) it suffices to take X = A ⊗LU U k and X ′

=

A′⊗

LU U k. �

Since over U , the Chevalley–Eilenberg complex V (n) of n constitutes a projective resolutionof U k, in Theorem 4.3 we concretely obtain X = A ⊗U V (n) = A ⊗k Λ∗n and X ′

=

A′⊗U V (n) = A′

⊗k Λ∗n, both equipped with the Chevalley–Eilenberg differential

d(a ⊗ y1 ∧ · · · ∧ yp) =

pi=1

(−1)i+1ayi ⊗ y1 ∧ · · · ∧ yi ∧ · · · ∧ yp

+

i< j

(−1)i+ j a ⊗ [yi , y j ] ∧ · · · ∧ yi · · · ∧ y j · · · ∧ yp

for a basis (yi )i for n. Let us look at some examples.If d = 1 or n = 1, we have n = kx1 ⊕ · · · ⊕ kxn , U = k[x1, . . . , xn], V (n) = K (x1, . . . , xn)

and X = K ( f1, . . . , fn). For n = 1 we have X ′= K ( f ′

1).Thus, the first non-trivial case to consider is d = 2 and n = 2. We have n = kx1 ⊕ kx2 ⊕

k[x1, x2] and consequently X is given by the complex

0 // Ad3

// A3d2

// A3d1

// A // 0

with basis elements over A given by x1 ∧ x2 ∧ [x1, x2] in degree 3, x1 ∧ x2, x2 ∧ [x1, x2],[x1, x2] ∧ x1 in degree 2, x1, x2, [x1, x2] in degree 1 and 1 in degree 0 and differentials given by

d3 =

[ f1, f2]

f1f2

, d2 =

− f2 0 [ f1, f2]

f1 −[ f1, f2] 0−1 f2 − f1

,

d1 =

f1 f2 [ f1, f2].

Similarly X ′ is given by the same expressions with A replaced by A′ and fi replaced by thechosen lift f ′

i . Note that [ f1, f2] = 0 but we possibly have [ f ′

1, f ′

2] = 0.

5. Deformations of schemes

In this section we specialize Theorem 3.8 to the scheme case. In Theorem 5.2, we give ageneral formulation in the setup of a Grothendieck deformation of the category Qch(X) over aquasi-compact, separated scheme. After discussing some special cases in which direct lifting ofKoszul complexes already leads to compact generation of the deformed category (like the casein which all deformed rings on an affine cover are commutative), in Section 5.4 we prove ourmain Theorem 5.10 which states that all non-commutative deformations are in fact compactlygenerated. The proof is based upon the change from Koszul complexes to liftable generatorsfrom Theorem 4.3.


5.1. Deformed schemes using ample line bundles

Let X be a quasi-compact separated scheme over a field k. If we want to investigate compactgeneration of D(D) for an abelian deformation D of C = Qch(X), by Proposition 3.7 (and infact, its special case [5, Proposition 5.9]) a global approach is to look for compact generators ofD(C) that lift to D(D) under k ⊗

LR −. We easily obtain the following result:

Proposition 5.1. Suppose X has an ample line bundle L. If H2(X,O X ) = 0, all infinitesimaldeformations of Qch(X) have compactly generated derived categories.

Proof. According to [18], D(Qch(X)) is compactly generated by the tensor powers Ln forn ∈ Z. By [12], the obstructions to lifting Ln along an infinitesimal deformation lie inExt2X (Ln, I ⊗k Ln) for I ∼= km for some m. But we have

Ext2X (L, I ⊗k L) ∼= [Ext2(L,L)]m= [Ext2X (O X ,O X )]

m= [H2(X,O X )]

m= 0

as desired. �

5.2. Deformed schemes using coverings

Let (X,O) be a quasi-compact, separated scheme and put C = Qch(X). Since thehomological condition in Proposition 5.1 excludes interesting schemes, we now investigate adifferent approach based upon affine covers.

Let Ui for i ∈ I = {1, . . . , n} be an affine cover of X , with Ui ∼= Spec(O(Ui )). PutZi = X \ Ui . With Ci = Qch(Ui ) and Si = QchZi (X), the category of quasi-coherent sheaveson X supported on Zi , we are in the situation of a covering collection of compatible localizationsof C.

For J ⊆ I , put UJ = ∩ j∈J U j and C J = Qch(UJ ). For i ∈ I and J ⊆ I \ {i}, putZ i

J = Ui \ ∪ j∈J U j = Ui ∩ ∩ j∈J Z j . The essential image E of ∩ j∈J S j −→ C −→ C/Si isgiven by E = QchZ i

J(Ui ).

Let ∆ and ∆∅ be as in Section 2.2. For K ⊆ J , we have UJ ⊆ UK and the correspondinglocalization is given by restriction of sheaves aK

J : Qch(UK ) −→ Qch(UJ ) with right adjointdirect image functor i K

J . Moreover, the localization can be entirely described in terms of modulecategories. If O(UK ) −→ O(UJ ) is the canonical restriction, then we have

aKJ

∼= O(UJ )⊗O(UK )− : Mod(O(UK )) −→ Mod(O(UJ ))

and the right adjoint i KJ is simply the restriction of scalars functor, which is obviously exact. For

the resulting pseudofunctor

Mod(O(U•)) : ∆ −→ Cat : J −→ Mod(O(UJ )),

we have Qch(X) ∼= Des(Mod(O(U•))). According to [13], this situation is preserved underdeformation. More precisely, up to equivalence an arbitrary abelian deformation ι : C −→ D isobtained as D ∼= Des(Mod(O•)) where

O• : ∆ −→ Rng : J −→ O J

is a pseudofunctor (a “twisted presheaf”) deforming O(U•), and D J = Mod(O J ) is thedeformation of C J corresponding to S J . By Proposition 2.6, the functors i K : DK −→ D areexact, and the ak : D −→ Dk constitute a covering collection of compatible localizations of D.


We note that by taking gi = {Oi }, condition (1) in Theorem 3.8 is automatically fulfilled. Weconclude:

Theorem 5.2. Let X be a quasi-compact, separated scheme with an affine cover Ui for i ∈ I =

{1, . . . , n}. Let ι : Qch(X) −→ D be an abelian deformation with induced deformations Diof Qch(Ui ). For every i ∈ I and J ⊆ I \ {i}, consider Z i

J = Ui ∩ ∩ j∈J Z j . Suppose thereis a collection gi

J of objects in D−(Di ) such that k ⊗LR gi

J compactly generates DZ iJ(Ui ) inside

D(Ui ). Then D(D) is compactly generated and an object in D(D) is compact if and only if itsimage in each D(Di ) is compact.

Remark 5.3. Before it makes sense to investigate the more general situation of deformationsof quasi-compact, quasi-separated schemes X , for which DQch(X)(Mod(X)) is known to becompactly generated by [2], a better understanding of the direct relation between deformationsof Qch(X) and Mod(X) should be obtained. It follows from [13] that these two Grothendieckcategories have equivalent deformation theories, the deformation equivalence passing throughtwisted non-commutative deformations of the structure sheaf. An interesting question in itsown right is to understand whether corresponding deformations of Qch(X) and of Mod(X)are related by an inclusion functor and a quasi-coherator like in the undeformed setup.

5.3. Twisted deformed schemes

In this section we collect some observations which follow immediately from Theorem 5.2,based upon direct lifting of Koszul complexes. In the slightly more restrictive deformation setupof Section 5.4, all compact generation results we state here also follow from the more generalTheorem 5.10, but there the involved generators are more complicated.

Let A be a commutative k-algebra and f = ( f1, . . . , fn) a finite sequence of elements in A.Put X = Spec(A). Consider the closed subset

Z = V ( f ) = V ( f1, . . . , fn) = {p ∈ Spec(A) | f1, . . . , fn ∈ p} ⊆ X.

Let QchZ (X) be the localizing subcategory of quasi-coherent sheaves on X supported on Z andput DZ (X) = DQchZ (X)(Qch(X)). We recall the following:

Proposition 5.4 ([1]). The category DZ (X) is compactly generated by K ( f ) inside D(X).

Let A be an R-deformation of A and let D = Mod(A) be the corresponding abeliandeformation of C = Qch(X) ∼= Mod(A). Let QchZ (X) ⊆ D be the localizing subcategorycorresponding to QchZ (X) ⊆ C. Let f = ( f1, . . . , fn) be a sequence of lifts of the elementsfi ∈ A to fi in A under the canonical map A −→ A. Clearly the precomplex K ( f ) of finite freeA-modules satisfies k ⊗R K ( f ) = K ( f ). If K ( f ) is a complex, we thus have

k ⊗LR K ( f ) = K ( f ).

From Proposition 3.7 we deduce:

Proposition 5.5. If every two distinct elements fl and fk commute, then K ( f ) compactlygenerates DQchZ (X)

(D) inside D(D).

Corollary 5.6. If A is a commutative deformation of A, then DQchZ (X)(D) is compactly

generated inside D(D).


Corollary 5.7. If f = ( f1) consists of a single element, then DQchZ (X)(D) is compactly

generated inside D(D).We can now formulate some corollaries of Theorem 5.2:

Proposition 5.8. Let X be a quasi-compact separated scheme with affine cover Ui for i ∈ I =

{1, . . . , n}. Let

O• : ∆ −→ Rng : J −→ O J

be a pseudofunctor deforming

O(U•) : ∆ −→ Rng : J −→ O(U•)

such that all the rings O J are commutative. Then the category D(D) for D = Des(Mod(O•))

is compactly generated and an object in D(D) is compact if and only if its image in each of thecategories Mod(Oi ) is compact.

In particular, we recover the fact that for a smooth scheme, the components H2(X,O X ) ⊕

H1(X, T X ) of H H2(X) correspond to compactly generated deformations of Qch(X), a factwhich also follows from [24].

Proposition 5.9. Let X be a scheme with an affine cover U1, U2 with Ui ∼= Spec(Ai ) such thatU1∩U2 ∼= Spec(A1x )

∼= Spec(A2 y) for x ∈ A1 and y ∈ A2. Then every deformation of Qch(X)is compactly generated.

Unfortunately, Proposition 5.9 typically applies to curves, and they tend to have no genuinelynon-commutative deformations. For instance, for a smooth curve X the Hochschild cohomologyis seen to reduce to H H2(X) = H1(X, T X ) for dimensional reasons, whence there are onlyscheme deformations of X .

5.4. Non-commutative deformed schemes

Let k be a field and (R,m) a finite dimensional k algebra with md= 0 and R/m = k. In this

section we prove our main result, namely that non-commutative deformations of quasi-compactseparated schemes are compactly generated. Based upon Section 4.3, we remedy the fact that forgeneral non-commutative deformations of schemes, the relevant Koszul precomplexes fail to becomplexes and hence cannot be used as lifts, unlike in the special cases discussed in Section 5.3.

Theorem 5.10. Let X be a quasi-compact separated k-scheme with an affine cover Ui fori ∈ I = {1, . . . , n}. Let ι : Qch(X) −→ D be an abelian R-deformation with induceddeformations Di of Qch(Ui ). Then D(D) is compactly generated and an object in D(D) iscompact if and only if its image in each D(Di ) is compact.

Proof. The theorem is an application of Theorem 3.8. For i ∈ I and J ⊆ I \ {i}, putY = Ui = Spec(A) and Z = Ui ∩ ∩ j∈J Z j . For a finite sequence of elements f = ( f1, . . . , fk)

we can write

Z = V ( f ) = {p ∈ Spec(A) | f1, . . . , fk ∈ p} ⊆ Y.

For the induced deformation Di of Qch(Ui ) ∼= Mod(A) we have Di ∼= Mod(A) for an R-deformation A of A. By Proposition 5.4, the category DZ (Y ) is compactly generated by K ( f )inside D(Y ). Now by Theorem 4.3, there exists a perfect complex X ′

∈ D(A) for whichA ⊗

LA

X ′= k ⊗

LR X ′ compactly generates DZ (Y ) inside D(Y ) ∼= D(A) as desired. �


The theorem shows in particular that the entire second Hochschild cohomology is realized bymeans of compactly generated abelian deformations.

Acknowledgments

The first author acknowledges the support of the European Union for the ERC grant No.257004-HHNcdMir. The second author is a director of research at the Fund for ScientificResearch, Flanders.

The first author thanks Tobias Dyckerhoff, Dmitry Kaledin, Bernhard Keller, AlexanderKuznetsov, Daniel Murfet, Amnon Neeman, Paul Smith, Greg Stevenson and Bertrand Toen forinteresting discussions on the topic of this paper. She is especially grateful to Bernhard Keller forhis valuable comments on a previous version of this paper. Both authors thank the two anonymousreferees for their corrections, comments and questions, in particular they are grateful to the firstreferee for pointing out Lemma 4.1.

Appendix. Removing obstructions

In this appendix we discuss an approach to removing obstructions to first order deformationsfrom [8] based upon the Hochschild complex, which applies in the case of length two Koszulcomplexes and thus leads to an alternative proof of Theorem 5.10 in the case of first orderdeformations of surfaces. We compare the explicit lifts we obtain in both approaches.

A.1. Hochschild complex

Let p be a k-linear abelian category. Recall that the Hochschild complex C(p) is the complexof k modules with, for n ≥ 0,

Cn(p) =

P0,...,Pn∈C

Homk(p(Pn−1, Pn)⊗ · · · ⊗ p(P0, P1), p(P0, Pn))

endowed with the familiar Hochschild differential. Let C(p) be the dg category of complexes ofp-objects with Hochschild complex C(C(p)) with

Cn(C(p)) =

C0,...,Cn∈C(p)

Homk(Hom(Cn−1,Cn)⊗ · · · ⊗ Hom(C0,C1),Hom(C0,Cn)).

An element of Cn(p) can be naively extended to C(p), yielding

Cn(p) −→ Cn(C(p)) : φ −→ φ.

A.2. Linear deformations and lifts of complexes

If m is the composition of the category p, then a Hochschild 2-cocycle φ corresponds to thefirst order deformation

(p = p[ϵ],m = m + φϵ).

Here Ob(p) ∼= Ob(p) and we denote objects in p by P for P ∈ p. For objects P0, P1 ∈ p, we havep(P1, P0) = p(P1, P0)[ϵ]. A morphism f : P1 −→ P0 in p naturally gives rise to a morphismf = f + 0ϵ : P1 −→ P0 in p(P1, P0), the trivial lift. For a complex (P, d) of p-objects, therethus arises a natural lifted precomplex (P, d) of p-objects.


In p we have

m(d, d) = φ(d, d)ϵ

and in fact,

[φ(d, d)] ∈ K (p)(P[−2], P)

is precisely the obstruction to the existence of a complex (P, d′) in K (p) with k ⊗R(P, d

′) ∼=

(P, d) in K (p) (see [12]). In general this obstruction will not vanish, but in some cases it is seento vanish on the nose.

Proposition A.1. If the differential d of P has no two consecutive non-zero components dn :

Pn −→ Pn−1, then φ(d, d) = 0 and (P, d) is a complex lifting (P, d).

A.3. Removing obstructions

If 0 = [φ(d, d)] ∈ Ext2p(P, P), following [8] we consider the morphism φ(d, d) : Σ−2 P −→

P and we turn to the related complex

P(1) = cone(φ(d, d)) = P ⊕ Σ−1 P

with differential

d(1) =

d φ(d, d)0 −d

.

The obstruction associated to the complex P(1) is then given by

φ(1) = φ(d(1), d(1)) =

φ(d, d) φ(d, φ(d, d))− φ(φ(d, d), d)

0 φ(d, d)

.

According to [8, Lemma 3.18], the degree two morphismφ(d, d) 0

0 φ(d, d)

: P(1) −→ P(1)

is nullhomotopic (a nullhomotopy is given by

0 01 0

).

Put ψ (1) = φ(d, φ(d, d)) − φ(φ(d, d), d). It follows that the obstruction associated to P(1)

is given by

φ(1) =

0 ψ (1)

0 0

∈ K (p)(P(1)[−2], P(1)).

In general there is no reason why φ(1) should be nullhomotopic, but in some cases it can beseen to be zero on the nose.

Proposition A.2. Suppose the differential d of P has no three consecutive non-zero componentsdn : Pn −→ Pn−1. Then ψ (1) = 0 and there is a complex P(1) ∈ K (p) with k ⊗R P(1) ∼= P(1) ∈

K (p).

Following [8, Proposition 3.16], we note that the original complex P can sometimes bereconstructed from P(1).


Proposition A.3 ([8]). If φ(d, d) is nilpotent, P can be constructed from P(1) using cones,shifts and direct summands. This applies in particular if d has no m consecutive non-zerocomponents dn : Pn −→ Pn−1 for some m ≥ 1.

Proof. This follows from the octahedral axiom (see [8, Proposition 3.16]). �

A.4. The case of Koszul complexes

The approach to removing obstructions discussed in Appendix A.3 applies to the case oflength two Koszul complexes. In this section we compare this approach with the solution fromSection 4.3. Let A be a commutative k-algebra with a first order deformation A determined by aHochschild 2 cocycle φ ∈ Homk(A ⊗k A, A). For a sequence f = ( f1, f2) of elements in A weconsider the Koszul complex (K ( f ), d) which is given by

0 // A f1

− f2

// A2f2 f1

// A // 0 .

In Appendix, we take p to be the category of finite free A-modules. The obstruction φ(d, d) :

Σ−2 K ( f ) −→ K ( f ) is determined by the element

α = φ( f1, f2)− φ( f2, f1) ∈ A.

The complex K ( f )(1) = cone(φ(d, d)) is given by

0 // Ad(1)3

// A3d(1)2

// A3d(1)1

// A // 0

with differentials given by

d(1)3 =

0− f1

f2

, d(1)2 =

− f2 0 0f1 0 0α f2 f1

, d(1)1 =− f1 − f2 0

.

Apart from the signs, the main difference with the complex X from Section 4.3 lies in the factthat here α depends on the Hochschild cocycle, whereas in X it is replaced by the constant value1. The nulhomotopy δ for the obstruction φ(1) gives rise to the lifted complex K ( f )(1)[ϵ] withdifferential d − δϵ. Concretely, the differential is given by

d(1)3 =

−ϵ

− f1f2

, d(1)2 =

− f2 0 −ϵ

f1 −ϵ 0α f2 f1

, d(1)1 =− f1 − f2 −ϵ

.

On the other hand, if for X ′ we choose f ′

i = fi + 0ϵ, then we have [ f ′

1, f ′

2] = αϵ and henceX ′ has differential d ′ given by

d ′

3 =

αϵf1f2

, d ′

2 =

− f2 0 αϵ

f1 −αϵ 0−1 f2 − f1

, d ′

1 =

f1 f2 αϵ.

Clearly, the computations leading to (d(1))2 = 0 and to d ′2= 0 are almost identical and have the

definition of α as main ingredient.


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